Effect of Compliant Walls on Secondary Instabilities in Boundary-Layer Transition Ronald D. Joslinj NASA Langley Research Center, Hampton, Virginia 23681 and Philip J. Morris J Pennsylvania State University, University Park, Pennsylvania 16802 Abstract For aerodynamic and hydrodynamic vehicles, it is highly desirable to reduce drag and noise levels. A reduction in drag leads to fuel savings. In particular for submersible vehicles, a decrease in noise levels inhibits detection. A suggested means to obtain these reduction goals is by delaying the transition from laminar to turbulent flow in external boundary layers. For hydrodynamic applications, a passive device which shows promise for transition delays is the compliant coating. In previous studies with a simple mechanical model representing the compliant wall, coatings were found that provided transition delays as predicted from the semi-empirical e" method. Those studies were concerned with the linear stage of transition where the instability of concern is referred to as the primary instability. For the flat-plate boundary layer, the Tollmien-Schlichting (TS) wave is the primary instability. In one of those studies, it was shown that three-dimensional (3-D) primary instabilities, or oblique waves, could dominate transition over the coatings considered. From the primary instability, the stretching and tilting of vorticity in the shear flow leads to a secondary instability mechanism. This has been theoretical described by Herbert based on Floquet theory. In the present study, Herbert's theory is used to predict the development of secondary instabilities over isotropic and non-isotropic compliant walls. Since oblique waves may be dominant over compliant walls, a secondary theory extention is made to allow for these 3-D primary instabilities. The effect of variations in primary amplitude, spanwise wavenumber, and Reynolds number on the secondary instabilities are examined. As in the rigid wall case, over compliant walls the subharmonic mode of secondary instability dominates for low-amplitude primary disturbances. Both isotropic and non-isotropic compliant walls lead to reduced secondary growth rates compared to the rigid wall results. For high frequencies, the non-isotropic wall suppresses the amplification of the secondary instabilities, while instabilities over the isotropic wall may grow with an explosive rate similar to the rigid wall results. For the more important lower frequencies, both isotropic and non-isotropic compliant walls suppress the amplification of secondary instabilities compared to the rigid wall results. The twofold major discovery and demonstration of the present investigation are: (1) the use of passive devices, such as compliant walls, can lead to significant reductions in the secondary instability growth rates and amplification; (2) suppressing the primary growth rates and subsequent amplification enable delays in the growth of the explosive secondary instability mechanism. 1. Introduction Research involving fiow over flexible walls was started in the late-1950's by Kramer^'^. Experimentally, Kramer found significant drag reductions using rubber coatings over rigid walls. Investigators in the 1960's fo- cused on the task of experimentally duplicating and the- oretically explaining Kramer's results. The majority of these studies failed to produce any comparable results; yet, the theoretical results laid the foundation for all fu- ture studies involving fiexible walls. Interest turned to- ward the use of compliant walls for turbulent drag reduc- tion. In the 1970's NASA^ and in the 1980's the Office of Naval Research'* sponsored investigations involving the use of compliant walls for the turbulent problem. Al- thought most of the results from this era were either in- conclusive or unsatisfactory, the contributions, together with earlier results, have acted as stepping stones to the understanding of the physically complex fiuid/wall interaction phenomena. A compre- hensive review of the pioneering studies was given by Bushnell, Hefner and Ash®, in particular for the tur- bulent fiow problem. More recent reviews were given by Riley, Gad-el-Hak and Metcalfe®, Gad-el-Hak^'®, and Carpenter^. Motivation for the present investigation is partially derived from the following favorable theoretical and ex- perimental results. In the early 1980's, Carpenter and Garrad**^'** showed theoretically that Kramer-type sur- faces could lead to potential delays in transition. Fur- ther, they indicated deficiencies in previous investiga- tions which may have prevented their achieving results comparable to Kramer's. Only recently, experiments performed by Willis*^ and Caster*^ showed favorable re- sults using compliant walls. As outlined in the above mentioned reviews, a number of investigations in the past ten years have been conducted involving flexible walls. A main emphasis of these studies was to under- stand the physical mechanisms involved in the fluid/wall interaction of transitional and turbulent flows. Most of these studies focused on the 2-D instability problem, ex- cept Yeo^'* who showed that a lower critical Reynolds number existed for the isotropic compliant wall for 3-D instability waves. Carpenter and Morris^® and Joslin, Morris and Carpenter^® have shown that 3-D Tollmien- Schlichting waves can have greater growth rates over compliant walls than 2-D waves. However, they showed that, even though 3-D waves may be dominant, transi- tion delays are still obtainable through the use of com- pliant walls. They considered a compliant wall model used by Grosskreutz^^ for his turbulent boundary-layer experiments. In this paper, the growth rates and ampliflcation of secondary instabilities over compliant walls are pre- dicted and compared to the rigid wall results. A the- ory for secondary instabilities is used which is based on Floquet Theory and was developed by Herbert^®. This theory agrees remarkably well with experiments, in par- ticular those of Klebanoff, Tidstrom and Sargent^^ for peak-valley splitting (fundamental) and Kachanov and Levchenko^*^ for peak- valley alignment (subharmonic). The secondary instability theory is extended to allow for 3-D primary instabilities, which are dominant over the compliant walls considered. Appropriate boundary con- ditions for the Grosskreutz^^ wall model are derived for the secondary analysis. In the next section, the primary instability problem is outlined. In section three, the secondary instability problem is discussed. The numerical methods, results, and a summary follow in the remaining sections. 2. Primary Wave Model Results for 2-D and 3-D primary instabilities over compliant and rigid walls have been presented by Joslin^"'^, Morris and Carpenter"'^®'"'^®. Since the secondary instability theory is based on and includes a primary in- stability analysis, brief derivations of the dynamic equa- tions and boundary conditions for the 3-D primary insta- bility problem are included below. The disturbances are represented as travelling waves which may grow or decay as they propagate. Nonlinear coupling is ignored so that individual components of the frequency spectrum may be studied. Additionally, the quasi-parallel assumption is made. Consider an incompressible laminar, boundary-layer flow over a smooth flat wall. The Navier-Stokes equa- tions govern the flow. The Blasius proflle is used to represent the mean flow. A small-amplitude disturbance is introduced into the laminar flow. A normal mode representation is given as {v[,Q[}(x, y, z, t) ={vi,Qi}(y) exp[i(a;a cos (f) +za sin (f) — ujt)] + c.c. (1) where vi and ili are the complex eigenfunctions of nor- mal velocity and vorticity, respectively. To obtain a real solution, complex conjugate solutions denoted by c.c. are required, a is the wavenumber, w is the frequency, and (f) is the wave angle. In general, a and w are complex lead- ing to an ambiguity in the system. For temporal analy- ses, a is a real specifled wavenumber and w is the com- plex eigenvalue. For spatial analyses, w is a real specifled frequency and a is the complex eigenvalue. For the com- pliant wall problem, Joslin, Morris and Carpenter^® have shown that the use of eqn. (1) leads to an overestimation of the growth of the wave as it propagates. The wave ac- tually propagates in a nearly streamwise direction which is in the direction of the group velocity, not normal to the wave fronts. In the present paper, the secondary insta- bilities are investigated using this simple representation of the primary instabilities. Since the present approach is conservative, it should exemplify the beneflts of using compliant walls as a means to obtain transition delays. Also, a major emphasis and motivation of the present study is to determine the behavior, or response, of the phenomena-namely secondary instabilities-to compliant walls. If the normal mode relation (1) is substituted into the linearized form of the Navier-Stokes equations, the following nondimensional system results: where v'{" + ai{y)v'{ + a2{y)vi = (2) ai(y) = — iRs{Uo{y)ci cos (j) — uj) — 2a a-ziy) = iRja^ {Uo{y)a cos (f) -Lu) +iRsa cos <f)Uo"{y) + a n'\ + a3(j/)fii + a4iy)vi = (3) asiy) =- a^ - iRs(Uo(y)a cos (f) - w) a-iiy) = - i Rs a sin (t)U ' o(y)- The equations are nondimensionalized using the freestream velocity Uoo, kinematic viscosity i>, and an appropriate length scale. Convenient lengths for the boundary-layer scale with the ^-Reynolds number, ijj, = Uoox/i'. These include a thickness, 6, where the Reynolds number is deflned Rs = Rx and a boundary-layer displacement thickness, 6* , where Rs- = where -I/O 17207i?j; . Eqns. (2) and (3) are referred to as the Orr- Sommerfeld and Squire equations, respectively. The sys- tem requires six boundary conditions. Requiring that the disturbance fluctuations vanish at infinity supplies three: vi{y) , v,'{y) , fii(j/)^0 y (4) The remaining boundary conditions are determined from the compliant wall model. The compliant wall model used for the present pa- per was introduced by Grosskreutz^^ in his experimental drag reduction studies with turbulent boundary layers. He suggested that the link between streamwise and nor- mal surface displacements would cause a negative pro- duction of turbulence near the wall. Although his results for the turbulent fiow were disappointing, the surface does react to the fiuid fiuctuations in transitional fiow in such a way as to reduce production of instability growth. Carpenter and Morris^^ have shown by an energy analy- sis how the many competing energy-transfer mechanisms are infiuenced by the compliant wall presence. Of note is the reduced energy production by the Reynolds stress which may cause the reduced growth rates. Further, Joslin, Morris and Carpenter^® predicted that transition delays of 4-10 times the rigid wall transition Reynolds number were achievable with this coating. So the model has been extended to allow for a secondary instability analysis. The mechanical model consists of a thin, elastic plate supported by hinged and sprung rigid members in- clined to the horizontal and facing upstream at an angle, 6, when in equilibrium. A sketch of the mechanical wall model is shown in Fig. 1. The boundary conditions are obtained by enforcing a balance of forces in the stream- wise and spanwise directions and the continuity of fiuid and wall motion. These are given below in linearized form. Fig. 1 Mechanical model representing the Grosskreutz compliant coating. For small displacements of an element out of equi- librium, the mechanical surface can be thought to move in a direction perpendicular to the rigid swivel-arm. The horizontal and vertical displacements (Ci,rii) are linked to the angular displacement (66) by ^1 = £,56' sin 6* and m £66 cos 6 (5) where £ is the length of the rigid-arm member. Equations of motion for the element in the streamwise and spanwise directions may be obtained by a balance of the forces of the fiuid fiuctuations acting on the surface and the forces due to the wall motion. These equations are dt^ dx^ 2Bx dx'^dz'^ +B, ^) cos2 6 + Ke m - E^h 5^6 {—P -\- Tyy) COS 6 + Tyx SlU 6 COS 6 and KsCi-E,b d\i sin 6 cos 6 (6) (7) where {Bx, Bz} {Ex,E,}b^ 12(1 - V^V;,) id Br yBxB;, Ci is the spanwise surface displacement, pm and h are the plate density and thickness; (B^, B^z, B^) are the fiexu- ral rigidities of the plate in the streamwise, transverse, and spanwise directions; {Ex, E^) are the moduli of elas- ticity of the plate; Ke,Ks are the effective streamwise and spanwise spring stiffness; p is the pressure fiuctua- tion which is obtained from the fiuid momentum equa- tions; and Tyx, Tyy and Ty^ are the streamwise, normal, and spanwise viscous shear stress fiuctuations in the fiuid acting on the wall. The terms on the left hand side of eqn. (6) refer to mechanical forces and the terms on the right refer to fiuid motion forces due to viscous stress and pressure fiuctuations. For the case where the ribs are aligned at 6 = Q° , the wall becomes isotropic and reduces to the theoretical model studied by Carpenter and Garrad^*^'^^. Otherwise the wall is referred to as non-isotropic and the rib angle is determined by 6. The continuity of fiuid/wall motion is given in the streamwise, normal, and spanwise directions, respec- tively as 56 dt drji dt dt — — = Ml +r]iUo Vl Wl (8) (9) (10) where (ui,vi,wi) are the disturbance velocity compo- nents in the streamwise, normal and spanwise directions. For the Grosskreutz coating, Ks ^ oo is assumed, which from eqn. (7) would result in zero effective spanwise surface displacement. This implies from eqn. (10) that wi(0) = 0. Strictly speaking, if the assumption Ks -^ co is relaxed, the resulting instabilities have larger growth rates. This suggests that spanwise stiffeners are stabiliz- ing to a disturbed flow. So with the assumption enforced, a better coating for potential transition delays results. The surface displacement takes the same normal mode form as the primary wave given by eqn. (1). The normal modes are substituted into eqns. (6-10). The equations can be reduced to three equations in terms of the normal velocity and vorticity-'^®'^"'^. 3. Secondary Instability Theory In this section, a secondary instability theory devel- oped by Herbert^® is extended to allow for 3-D primary instabilities. Additionally, boundary equations describ- ing the compliant walls are introduced for secondary in- stabilities. The flow is governed by the Navier-Stokes equations. Instantaneous velocity and pressure compo- nents are introduced and given as v(x,y,z,t) = V2(x,y,z,t) + Bv3(x,y,z,t) p{x, y, z, t) = p'zix, y, z, t) -^ Bpz{x, y, z, t) (11) where ps and v^ = (m3,i'3,W3) are the secondary dis- turbance pressure and velocity in the fixed laboratory reference frame (x,y,z); and p2 and V2 = (m2,^2,m'2) are the basic pressure and velocity given by, V2{x,y,z,t) = {Uo{y),0,0} + A{ui,vi,wi}{x,y,z,t) P2{x,y,z,t) = Api{x,y,z,t) (12) The basic fiow is given by the Blasius profile and eigen- functions of the primary wave. Assume locally that the primary wave is periodic in t and periodic in {x, z) with wavelength A^ = 2t: / a^ and efine a disturbance phase velocity which is Cr = {Cx = LOr I Ctr COS (/), 0, C^ = LO^ / Ctr sin (j)) . Then in a frame moving with the primary wave, vi{x, y, z) = vi(x, y, z) = vi(x + \x,y, z + X;,) (13) where (x, z) is the reference frame moving with the wave. With an appropriate normalization of primary eigen- functions {ui,vi,wi) the amplitude. A, directly mea- sures the maximum streamwise rms fiuctuation. This is given by max |«i(j/)p = |«i(j/™)P = 1/2 (14) 0<y <oo The instantaneous velocities and pressure (11) are substituted into the Navier-Stokes equations which are linearized with respect to the secondary amplitude, B. The disturbance pressure is eliminated, resulting in the vorticity equations. -^V2 - 1-1 n3-{v2 ■ V)fi3 - (V3 ■ V)fi2 Kg Ot i + (^2 • V)V3 + (^3 • V)V2 = (15) with the continuity equation V -173 = (16) As with the primary problem, the equations are formed in terms of the normal velocity (^3) and vorticity {^3). The equations are found to take the form^^, [7^V2-^-(t^o-C.)f + C A]fi3 ^Rj ot ox OZ-' + A{-(Fi-V)fi3-(i73-V)fii dUo dv3 dy dz + ( _|_ ^^^ _|_Q ^_|_ r _ ^^^ dz dx dy dx dz dvi du3 dvi dw3 ~i dz dy dx dy J (17) and, iir^ -7^-iUo-c.)T- + c,T-]v'v3■^ - Rs d_ dt d_ d_ ' dx dz" dy"^ dx I dy dx'-^ dz'-^ dy'-' ^d'^vi. d ,d'^vi d'^vi d'^vi „ 9^mi , d _l_ 2 —) 1- ( - -| - -\- 2 —) — dxdy dx dz'^ dx^ dy^ dxdy dy .d'^wi d'^wi d'^ui d'^vi . d ,dvi „9mi, _|_ ( i i _| i —\ 1_ ( — 1 _|_ 2 — -) dy'^ dx^ dxdz dydz dz dy dx d^ d^ d^ dvi d^ r)/9ui dwi d^ dz"^ dx"^ dy"^ dx dxdy dz dx dxdz dvi d"^ 1 ^ r_ A^v^t; ) ^ 2(^^ - ^^ -F ^^ dz dydz dx dz"^ dx"^ dy"^ d'^ui d ^.d'^wi d'^vi d dvi d^ d^ dxdy dx dxdy dxdz dz dx dx'-' dz'-' dy'-' dy dx dxdy dx dydz dvi 9^ 1 r 9 2 \ o/'^^"! "^^^i \ ^ dz dxdz dz dydz dxdz dx dvi , d^ 9^ , „9mi d^ dz dx'' dy'' dz dxdy ]W3} (18) The disturbance quantities V3, Q3 and dv3/dy are re- quired to vanish far from the wall and at the wall for the rigid wall case. The compliant wall equations give the re- maining boundary conditions in the compliant case. Ad- ditionally, the primary amplitude, A, is a parameter in the equations and is assumed to be locally non-varying. As j4 ^ the Orr-Sommerfeld and Squire equations re- sult. For the case of interest where A^ 0, the primary eigenfunctions (mi, vi, wi) appear in the equations as co- efficients. To solve the secondary problem, a normal mode so- lution having the following form is assumed, V3{x,y,z,t) = e^'+'f^^' ""^ "^-^ "''' "^Wix, y, z) (19) where /3 = 2t: /X^, is a specified spanwise wavenumber and (T = (Tr -|- iiTi is a temporal eigenvalue or is spec- ified for spatial analyses. V(x,y,z) is a function that represents the class of secondary modes. Floquet theory suggests the form of solution for periodic systems. For the present problem, this may be written V(x, y, z) = eT(^ ''°'"f'+' """^ 'f''^V(x, y, z) (20) where 7 = 7^ + ^Ti is the characteristic exponent and V{x, y, z) is periodic in the {x, z) plane and may be rep- resented by a Fourier series. Thus the representation of the secondary instability for a 3-D basic fiow is. vs =e i7t-\-il3(z cos (f) — x sin (f)')-\-^(x cos (f)-\-z sin (/>) E ''-^yy^" l2)ar{x cos (j)-\-z sin (/>) (21) n = — 00 This suggests a form of solution for the secondary distur- bance based on a coordinate system oriented at an angle 4> with respect to the mean fiow and moving with the primary wave. If the coordinate system is aligned with the primary wave, or (/) = 0°, then the solution for the secondary disturbance would follow Herbert, Bertolotti and Santos^^ who considered a 2-D primary wave. If solutions given by eqn. (21) are substituted into eqns. (17) and (18), an infinite system of ordinary differ- ential equations result. The dynamic equations are de- termined by collecting terms in the governing equations with like exponentials. The system consists of two dis- tinct classes of solution because the even and odd modes decouple. Even modes correspond to the fundamental mode of secondary instability, and the odd modes are the subharmonic mode. Only a few terms of the Fourier series are retained since, as shown by Herbert, Bertolotti and Santos^^, this provides a sufficiently accurate ap- proximation for a 2-D disturbance. This form of solution indicates two complex quanti- ties, (T and 7, which leads to an ambiguity similar to that found with the Orr-Sommerfeld/Squire problem. There are four unknowns, <Tr,<^i,lr,li- Two can be de- termined while two must be chosen in some other way. In the present study, temporally-growing tuned modes are examined. The temporal growth rate is cr^, and <Ti can be interpreted as a shift in frequency. In this case, 7^ = 7i = 0. If (Tj- = 0, then the secondary disturbance is travelling synchronously with the basic fiow. The boundary conditions for the secondary distur- bance are given as Vn,v',^n ^0 y (22) along with the compliant wall boundary conditions. In the rigid wall case. ,f^n=0 at y (23) The analysis for the compliant boundary conditions for secondary instabilities follows the same route as was taken for the primary instabilities, except a number of additional terms arise due to the presence of the primary wave. The fiuid/wall motion must be continuous in each direction. In addition, the equations of force (6, 7) must balance in the streamwise and spanwise directions in the reference frame moving with the primary wave. Con- sistent with the fiuid equations, the amplitude of the primary wave is assumed to be locally non-varying. In deriving the final form of the wall equations, a signifi- cant difference between the primary and secondary form arises from the pressure contribution. The pressure for the secondary disturbance is determined from the mo- mentum equations which are complicated by primary coupling terms. The continuity of motion between the fiuid and solid is given by 56 dt -- «3 + mK + ^{(6 • v)«3 + (6 • v)«i} (24) ^ = ^3 + a[{1, ■ V)vs + (6 • V)t;i } (25) ^ = ws + A[{i,-V)ws + {i^-V)w,] (26) where -: „ ^ d d ^ d ox ay oz Equations (24-26) involve six unknowns for the velocity fiuctuation and surface displacement in a highly coupled system. As with the primary boundary conditions, it is possible to derive a set of equations which represent the surface motion in terms of the normal velocity and vorticity only. This is algebraically very tedious. A com- plete derivation is given by Joslin^"'^. Note that if A = in the secondary wall equations, the primary wall equa- tions result. This occurred with the fiuid equations as well. 4. Numerical Methods of Solution The algebraic complexity of the dynamic equations for the secondary disturbance and the compliant wall equations requires that care be taken in applying any numerical technique. Because no theoretical or exper- imental data are available for the compliant problem, both shooting^® and spectraP^ approximations are used. Also, noting that Bertolotti^'* has shown for the rigid wall problem with a 2-D primary instability that, after the transformation from spatial to temporal, the solu- tions are in good agreement, a temporal analysis is pre- sented in this paper. For the spectral method, Chebyshev series are intro- duced to approximate each mode of the Fourier series. An algebraic transformation is used to change from the Chebyshev spectral domain [—1, 1] to the physical do- main. Due to the properties of the Chebyshev polyno- mial, the equations are recast in integral form. Cheby- shev polynomials are used to represent the basic flow in the series which are substituted into the integral equa- tions. For the basic flow, 35 polynomials provided suffi- cient resolution of the eigenfunctions. The series repre- senting the secondary instability requires 40 polynomi- als for sufficient convergence to the dominant eigenvalue. For the shooting method, beginning with the equations for the compliant wall, integrations of the disturbance equations across the boundary layer are performed us- ing a Runge-Kutta scheme. At the edge of the bound- ary layer, the numerical solution vectors are matched with the asymptotic solutions. A very accurate initial guess is found to be required for convergence using this method. To demonstrate the accuracy of the numerical techniques, a comparison for the rigid wall case is made with Herbert^s for Rs = 826.36, Fr = 83, /3 = 0.18, and A = 0.02. Herbert obtained the dominant mode a = 0.01184. In good agreement, the present spec- tral and shooting methods lead to cr = 0.011825 and a = 0.011839, respectively. 5. Results For all of the results that follow, the freestream ve- locity is 20 m/s, the density is 1000 kg/m^, and the kine- matic viscosity is 1 x 10~® m^/s. The coatings considered consist of both isotropic and non-isotropic walls. Both walls were optimized at Rs- = 2240 for 2-D primary instabilities. The isotropic wall has properties 9 = 0°, h = 0.735mm, E^ = 1.385MN/m2, K = 0.354GN/m3 and Pfn = 1000 kg/m^; and the non-isotropic wall has properties = 60°, h = 0.111mm, E^ = 0.509MN/m2, K = 0.059GN/m3 and pm = 1000 kg/m^. A Reynolds number of 2240 was chosen because, for a boundary layer over a rigid wall, the disturbance with the critical fre- quency (in the e" sense) reaches its maximum growth rate near this value of Reynolds number. Accordingly, this is a good choice of Reynolds number for optimizing the wall properties. In considering 3-D instabilities, the walls optimized for 2-D instabilities are used with the ad- dition of isotropic plates. The properties of an isotropic plate are direction independent; that is, E^ = E^,. Al- though complete details of the optimization process and philosophy are given by Carpenter and Morris^^, a recap follows. With a flexible wall present, other modes of instabil- ity arise. With changes in the compliant wall properties, stable, or marginally stable, fluid and wall modes can be- come unstable and dominant. The present wall proper- ties were varied to achieve an optimal specifled condition. This desired condition was to achieve a minimum growth rate for a dominant 2-D Tollmien-Schlichting instability while keeping other modes marginally stable. For the secondary analysis, these "optimal" compliant walls led to no additional unstable modes. However, this is not to say that additional growing modes may not appear for different wall properties. In this analysis, the primary wave amplitude {A) and the secondary instability spanwise wavenumber (/3) are parameters of the problem. Herbert^® showed in his boundary-layer studies for the 2-D primary wave over a rigid wall that as the amplitude increases the growth rate of the secondary instability increases. Also, as the span- wise wavenumber is varied, the temporal growth rate reaches a maximum for a particular wavenumber. Ad- ditionally, Herbert showed that the subharmonic mode reaches greater growth rates than the fundamental mode for low amplitude disturbances. These flndings were verifled by the direct numerical simulations of Spalart and Yang^^. Althought both the subharmonic and fun- damental modes over the compliant walls were exam- ined, emphasis is placed on the subharmonic mode, since as both theory and computations indicate, subharmonic disturbances are more unstable than fundamental distur- bances for small amplitudes. Limited fundamental dis- turbance results are included to verify that these modes do not become the dominant instability over compliant walls. Primary waves with frequencies that give maxi- mum disturbance growth rates are considered. For the isotropic wall, the maximum growth rate occurs at a fre- quency Lu = 0.065 (Fr =~ 29.0), where Fr = w/Rx 10'^. Figure 2 shows the growth rates of the subharmonic and fundamental disturbances as a function of the spanwise wavenumber for the rigid wall and isotropic compliant wall. As the flgure shows, growth rates over the com- pliant wall are reduced in comparison with the rigid wall results over the whole range of spanwise wavenum- bers. Additionally, the subharmonic disturbance has much larger growth rates than the fundamental distur- bance, as expected. Similar trends are found in the com- parison of non-isotropic and rigid wall results. In consid- ering reductions in the growth rates of the subharmonic mode as a result of compliant walls, the isotropic wall suppressed the maximum growth rate by 20%. For the maximum growth rate over the non-isotropic wall, which occurs at a difference frequency than the isotropic wall case, the non-isotropic wall led to a reduction of 17% compared to rigid wall results. So for a fixed Reynolds number and primary wave amplitude, both isotropic and non-isotropic compliant walls lead to reduced secondary instability growth rates compared to the rigid wall re- sults. .010 J- f i .IXID d 1 3 S^UbIH WUHBUnbd , & Fig. 2 Growth rates of the secondary instabil- ities as a function of spanwise wavenuniber for Rs, = 2240, Fr ~ 29.0, and A = 0.01. subharmonic: — o— , rigid wall; • • o- • •, isotropic wall and funda- mental: — X — , rigid wall; • • • x • • •, isotropic wall. A more revealing measure of the effectiveness of us- ing compliant walls to suppress secondary instabilities is to compute the amplitude growth and decay with down- stream distance. The amplification of the primary and secondary instabilities are governed by of waves at a frequency Fr = 53. At this frequency, primary amplitudes over the the isotropic wall are sim- ilar to those over the rigid wall. This suggests that the development of secondary instabilities might also be sim- ilar, since secondary disturbances are parametrically de- pendent on the basic fiow. For the non-isotropic wall results shown in Figure 4, the primary amplitudes are suppressed significantly compared to the rigid wall and isotropic wall cases. Therefore, the non-isotropic wall would likely lead to a very different secondary instability development, most probably with reduced amplitudes. Again from Figure 4, one might expect greater differ- ences in the secondary instability development over both compliant walls as the Reynolds numbers increase and corresponding frequencies decrease. > |- n r m~ _I_L j-j- I I I I I I _I_L 3W3 <oo sw *w 700 Fig. 3 Amplitude growth as a function of Reynolds number for the subharmonic mode (B) of a 2-D primary wave (A) over a rigid wall at Fr= 124, Ao = 0.0044, 5o = 1.86x 10-^ and & = 0.33. , theory and (x,o), Kachanov and Levchenko [15]. ■4 t* dx ■■4 —^dx (27) where Ag, Bg are the initial amplitudes aX Xg, A and B are the amplitudes at a downstream distance x, and * denotes dimensional quantities. As shown by Herbert^®, the theoretical prediction of primary and secondary am- plification by eqns. (27) compares well with the exper- iments of Kachanov and Levchenko^*^. A similar com- parison is shown in Fig. 3, where the theoretical results were obtained with the present numerical techniques. Before computing similar amplification results over the compliant walls, inferences of the secondary insta- bility growth may be drawn from primary instability results. Figure 4 shows the maximum amplification of various frequency primary waves propagating over the rigid and compliant walls along with the amplification J J L. J L 0.0 ^ ^-^^^ hiCTmiUc munbcT, Rr ^ U*"^ 1.5 Fig. 4 2-D curves of maximum amplification for TSI waves over a , rigid wall; , isotropic wall; and , non-isotropic wall and — • — , waves of Fr ~ 53. To demonstrate these postulations for secondary in- stabilities, eqns. (27) are used to compute the amplifi- cation of primary and secondary instabilities over the rigid, isotropic, and non-isotropic walls for Ag = 0.004, Bo = lx 10-^ and h = 0.15, where h = 13/ R x 10^. The initial amplitudes are somewhat arbitrary: the present values were selected to be close to the experiments of Kachanov and Levchenko^*^ for rigid walls. The spanwise wavenumber (h) was chosen near the maximum growth rate of the secondary instability at the branches of the neutral curve for the rigid wall case. Both primary and secondary amplifications are shown in Fig. 5. Clearly, the results of the secondary instability growth over the isotropic and rigid walls are similar, as postulated, while the non-isotropic wall significantly suppresses the sec- ondary instability growth. 9 I ft-*' - 4|i.D2S ■M CI . n7 :< lUymifali:] Fig. 5 Amplitude growth as a function of Reynolds number for the subharmonic mode (B) of a 2-D primary wave (A) at Fr ~ 53, Ag = 0.004, Bo = I X 10~®, and h = 0.15 over a , rigid wall; , isotropic wall; and , non-isotropic wall. The growth of the secondary instability is dependent on the parameters of the basic fiow, most probably the primary instability amplitude (A). For example. Fig. 6 shows the amplification of the primary and secondary instabilities over the rigid wall with properties as before and over the non-isotropic wall with both Ag = 0.004 and Ag = 0.008. Even by doubling the initial amplitude of the primary disturbance, the growth of the secondary instability over the non-isotropic wall continues to be suppressed and has not exceeded the primary amplitude upon crossing the neutral curve. Yet, the doubled initial amplitude (Ag = 0.008) results in a significant increase in the secondary instability growth compared with the lower amplitude (Ag = 0.004) results. Hence, the sup- pression of the primary instability amplitude is of utmost importance to suppress the onset of the secondary insta- bility growth. 0,0* Q.-DG CDS 0.10 0.12 14 Fig. 6 Amplitude growth as a function of Reynolds number for the subharmonic mode (B) of a 2-D primary wave (A) at Fr ~ 53, Bg = 1 X 10~®, and b = 0.15 over a , rigid wall with Ag = 0.004; — • — , non-isotropic wall with Ag = 0.004; and , non-isotropic wall with Ag = 0.008. .MS I- .om - .002 - .000 Ifjiimr •iTrnnmh rii P Fig. 7 Growth rates of the subharmonic distur- bance for 2-D and 3-D primary waves over the isotropic wall as a function of spanwise wavenum- ber for Rs' = 2240, Fr ~ 22.3, and A = 0.01. , <j, = 0«; ■■■,<j> = 10°; and , <j> = 20°. Proceeding with investigating the effect of compli- ant walls on secondary instabilities, 3-D primary waves are introduced and are determined by the specified wave angle ((f)). In Fig. 7, subharmonic disturbance growth rates over the isotropic compliant wall are shown with variation in spanwise wavenumber and primary wave an- gle ((/)). A frequency w = 0.05 (Fr ~ 22.3) is selected since the 3-D primary wave growth rate is maximized at this frequency for the Reynolds number Rs- = 2240. Subharmonic growth rates arising from the 2-D primary wave ((f) = 0°) are clearly larger than those from the 3-D waves. As the primary wave angle ((f)) increases, the subharmonic growth rates continually become more damped. Additionally, for oblique waves ((f) ^ 0°) the secondary disturbances no longer travel synchronously with the primary wave. This is shown in Fig. 8 by the fre- quency shifts that result over the isotropic wall. Similar results occur for the non-isotropic wall. It is likely that this shift leads to a reduced efficiency of energy transfer from the basic flow to the secondary disturbance. This frequency shift is as much a result of the 3-D nature of the basic flow as it is of the compliant wall influence. Re- sults similar to those of Fig. 8 were found by Balachan- dar, Streett and Malik^® for the rotating-disk problem which also has a 3-D basic flow. 0.00 I- ■Q.O? - Q.Q* ' m _ • X "- . \ -. - \ ' \ ■ \ . , 1 . . , , 1 . , , , 1 Q.O 5.1 0.2 9 e,j Fig. 8 Frequency shift of the subharmonic distur- bance for 3-D primary waves over the isotropic wall as a function of spanwise wavenuniber for Rs, = 2240, Fr ~ 22.3, and A = 0.01. ■■■,<j> = 10° and , <j> = 20°. The secondary growth rate comparison in Fig. 7 is misleading as a result of holding the primary amplitude (A) flxed. With a variation in wave angle ((f)), corre- sponding primary wave amplitudes result as shown by Joslin, Morris and Carpenter^®. Then a more realistic comparison of secondary growth rates arising from 2-D and 3-D primary waves should involve amplitudes suited to a given wave angle ((f)). One means to derive these amplitudes is through the use of eqns. (27). In Fig. 9, a comparison of the subharmonic growth rates over the isotropic compliant wall is made for 2-D and 3-D primary waves allowing for amplitude differences. Similar results are found for the non-isotropic wall as well. As shown, the 3-D primary wave does lead to much larger growth rates than a 2-D wave for the compliant walls considered. This clearly demonstrates that in spite of the loss of syn- chronization with the basic flow, 3-D primary waves lead to dominant secondary instabilities over compliant walls. Again, the amplitude of the primary wave is of utmost importance for determining the secondary disturbance growth. Yet, compliant walls do lead to a reduction in secondary growth rates compared with those for the rigid wall. Also, recall that the amplitudes for 3-D primary waves used in these calculations were determined by a normal mode assumption which leads to a conservative estimation of the primary amplitude^®, and a more re- alistic lower amplitude would lead to secondary growth rates somewhere between the 2-D and 3-D results shown in Fig. 9. But, the goal here is to determine the fun- damental effect of compliant walls on secondary distur- bances. Final amplitude calculations using eqns. (27) are carried out for the frequency Fr ~ 53. In Fig. 10, sec- ondary amplitudes arising from the most amplifled 3- D primary wave over a non-isotropic compliant wall are compared to the previous results from 2-D primary waves over rigid and non-isotropic walls. At this frequency, the amplitude difference between the 2-D and 3-D primary instabilites is small, yet the secondary disturbance re- sponds notably. This is an indication that small changes in the primary instability, however slight, have a mount- ing effect on the rapidly developing secondary instability. ,0490 i.onss .oooa a.o a. 2 i>.i 0,4 Fig. 9 Growth rates of the subharmonic distur- bance for 2-D and 3-D primary waves over the isotropic wall as a function of spanwise wavenuni- ber for Rs' = 1760 and Fr ~ 30.2 for , <j> = 0" with A = 0.010; • • • , ,?i = 45° with A = 0.031. I |- 5*"-^ - 1 ^ .^^ fr p.na n.iH o.oa O.OJt o.ia O.IS D.I4 Fig. 10 Amplitude growth as a function of Reynolds number for the subharmonic mode (B) of a 2-D and 3-D primary wave (A) at Fr ~ 53, Bo = lx 10-^ Ao = 0.004, and h = 0.15 over a , rigid wall — • — , 2-D non-isotropic wall; and • • •, 3-D non-isotropic wall. 6. Summary In earlier studies^®'^®'^^, it was shown that 3-D pri- mary instabilities theoretically dominate transition over the compliant walls considered, yet transition delays were found compared to the rigid wall. The present pa- per has further extended the understanding of the effect of compliant walls on transition mechanisms in bound- ary layers. Namely, the effect compliant walls have on secondary instabilities has been investigated. It has been shown that the use of compliant walls can lead to reduced growth rates and amplification of secondary instabili- ties. From both the earlier 3-D primary results combined with the present secondary findings, it should be empha- sized that the physical nature and make-up of the mech- anisms in transition are not altered by the control device (i.e. compliant wall). Rather, only the response of that mechanism is changed. This fact is of particular impor- tance for designing Laminar Flow Control (LFC) studies. As an example, the behavior of a secondary instability growth with variation in primary amplitude is well doc- umented by Herbert. As the primary amplitudes are reduced, the excitement of the secondary instability is delayed. Thus, active or passive devices which suppress primary instability growth should lead to corresponding suppression and delay of succeeding instabilities. This has been demonstrated above with the compliant wall. The twofold major discovery and demonstration of the present investigation is: (1) the use of passive devices, such as compliant walls, lead to significant reductions in the secondary instability growth rates and amplification; (2) suppressing the primary growth rates and subsequent amplification enable delays in the growth of the explosive secondary instability mechanism. 7. Acknowledgments Support for this project was supplied by the Naval Sea Systems Command and the Applied Research Labo- ratory Exploratory and Foundational Research Program under NAVSEA N0039-88-C-0051 at the Department of Aerospace Engineering, The Pennsylvania State Univer- sity. A grant for computational support was provided by the National Science Foundation with the Pittsburgh Supercomputing Center. 8. 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